Large-Scale Fading Coefficient Estimation in Wireless Massive MIMO Systems

ABSTRACT

Efficient algorithms for estimating LSFCs with no aid of SSFCs by taking advantage of the channel hardening effect and large spatial samples available to a massive MIMO base station (BS) are proposed. The LSFC estimates are of low computational complexity and require relatively small training overhead. In the uplink direction, mobile stations (MSs) transmit orthogonal uplink pilots for the serving BS to estimate LSFCs. In the downlink direction, the BS transmits either pilot signal or data signal intended to the MSs that have already established time and frequency synchronization. The proposed uplink and downlink LSFC estimators are unbiased and asymptotically optimal as the number of BS antennas tends to infinity.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. §119 from U.S.Provisional Application No. 61/904,076, entitled “Large-Scale FadingCoefficient Estimation in Wireless Massive MIMO Systems,” filed on Nov.14, 2013, the subject matter of which is incorporated herein byreference.

TECHNICAL FIELD

The disclosed embodiments relate generally to wireless networkcommunications, and, more particularly, to large-scale fadingcoefficient estimation for wireless massive multi-user multiple-inputmultiple-output (MU-MIMO) systems.

BACKGROUND

A cellular mobile communication network in which each serving basestation (BS) is equipped with an M-antenna array, is referred to as alarge-scale multiuser multiple-input multiple-output (MIMO) system or amassive MIMO system if M>>1 and M>>K, where K is the number of activeuser antennas within its serving area. A massive MIMO system has thepotential of achieving transmission rate much higher than those offeredby current cellular systems with enhanced reliability and drasticallyimproved power efficiency. It takes advantage of the so-calledchannel-hardening effect that implies that the channel vectors seen bydifferent users tend to be mutually orthogonal andfrequency-independent. As a result, linear receiver is almost optimal inthe uplink and simple multiuser pre-coders are sufficient to guaranteesatisfactory downlink performance.

To achieve such performance, channel state information (CSI) is neededfor a variety of link adaptation applications such as precoder,modulation and coding scheme (SCM) selection. CSI in general includeslarge-scale fading coefficients (LSFCs) and small-scale fadingcoefficients (SSFCs). LFCSs summarize the pathloss and shadowingeffects, which are proportional to the average received-signal-strength(RSS) and are useful in power control, location estimation, handoverprotocol, and other application. SSFCs, on the other hand, characterizethe rapid amplitude fluctuations of the received signal. While allexisting MIMO channel estimation focus on the estimation of the SSFCsand either ignore or assume perfect known LFCSs, it is desirable to knowSSFCs and LSFCs separately. This is because LSFCs can not only be usedfor the aforementioned applications, but also be used for the accurateestimation of SSFCs.

LSFCs are long-term statistics whose estimation is often moretime-consuming than SSFCs estimation. Conventional MIMO CSI estimationusually assume perfect LSFC information and deal solely with SSFCs. Forco-located MIMO systems, it is reasonable to assume that thecorresponding LSFCs remain constant across all spatial sub-channels andthe SSFC estimation can sometime be obtained without the LSFCinformation. Such assumption is no longer valid in a multiuser MIMOsystem, where the user-BS distances spread over a large range and theSSFCs cannot be derived without the knowledge of LSFCs.

In the past, the estimation of LSFC has been largely neglected, assumingsomehow perfectly known prior to SSFC estimation. When one needs toobtain a joint LSFC and SSFC estimate, the minimum mean square error(MMSE) or least squares (LS) criterion is not directly applicable. Theexpectation-maximization (EM) approach is a feasible alternate but itrequires high computational complexity and convergence is notguaranteed. A solution for efficiently estimating LSFCs with no aid ofSSFCs is sought in a massive multiuser MIMO system.

SUMMARY

Efficient algorithms for estimating LSFCs with no aid of SSFCs by takingadvantage of the channel hardening effect and large spatial samplesavailable to a massive MIMO base station (BS) are proposed. The LSFCestimates are of low computational complexity and require relativelysmall training overhead. In the uplink direction, mobile stations (MSs)transmit orthogonal uplink pilots for the serving BS to estimate LSFCs.In the downlink direction, the BS transmits either pilot signal or datasignal intended to the MSs that have already established time domain andfrequency domain synchronization. The proposed uplink and downlink LSFCestimators are unbiased and asymptotically optimal as the number of BSantennas tends to infinity.

In one embodiment, a base station (BS) receives radio signalstransmitted from K mobile stations (MSs) in a massive MIMO uplinkchannel where M>>K. The BS vectorizes the received radio signals denotedas a matrix Yε

^(M×T). The transmitted radio signals are orthogonal pilot signalsdenoted as a matrix Pε

^(K×T) transmitted from the K MSs, and T≧K is the pilot signal length.The BS derives an estimator of large-scale fading coefficients (LSFCs)of the uplink channel without knowing small-scale fading coefficients(SSFCs) of the uplink channel. The BS may also receive pilot signalsthat are transmitted for J times over coherent radio resource blocksfrom the K MS. The BS then derives a more accurate estimator of theLSFCs of the uplink channel based on the multiple pilot transmissions.In addition, the BS calculates element-wise expression of the LSFCs foreach of the kth uplink channel based on the LSFCs estimator.

In another embodiment, a mobile station (MS) receives radio signalstransmitted from a base station (BS) having M antennas in a massive MIMOsystem. The transmitted radio signals are denoted as a matrix Qtransmitted from the BS to K MS and M>>K. The MS determines a receivedradio signal denoted as a vector x_(k) received by the MS that is thekth MS associated with a k^(th) downlink channel. The k^(th) MS derivesan estimator of a large-scale fading coefficient (LSFC) of the k^(th)downlink channel without knowing a small-scale fading coefficient (SSFC)of the kth downlink channel. In a semi-blind LSFC estimation, matrix Qis a semi-unitary matrix consisting of orthogonal pilot signals, and theLSFC of the k^(th) downlink channel is derived based on x_(k) and thetransmitting power of the pilot signals. In a blind LSFC estimation,matrix Q represents pre-coded data signals transmitted to K′ MS that aredifferent from the K MS. The LSFC of the k^(th) downlink channel isderived based on x_(k) and the transmitting power of the data signalswith unknown data information and unknown beamforming or precodinginformation.

Other embodiments and advantages are described in the detaileddescription below. This summary does not purport to define theinvention. The invention is defined by the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates simplified block diagrams of a base station and aplurality of mobile stations in a single-cell massive MU-MIMO system inaccordance with one novel aspect.

FIG. 2 is an exemplary diagram illustrating an uplink MIMO system inaccordance with one novel aspect.

FIG. 3 is a flow chart of an uplink LSFC estimator that estimates allaccessing MSs' LSFCs simultaneously in a massive MIMO system.

FIG. 4A is a flow chart of an uplink LSFC estimator that estimates theLSFC for each accessing MS individually in a massive MIMO system.

FIG. 4B is a flow chart of an uplink LSFC estimator that estimates theLSFC for each accessing MS individually using a row of a diagonal matrixas pilot in a massive MIMO system.

FIG. 5 shows an exemplary schematic view of an uplink LSFC estimator ina massive MIMO system.

FIG. 6 is a flow chart of an uplink LSFC estimator that estimates allaccessing MSs' LSFCs simultaneously with multiple pilot transmissions ina massive MIMO system.

FIG. 7A is a flow chart of an uplink LSFC estimator that individuallyestimates the LSFC for each MS with multiple pilot transmissions in amassive MIMO system.

FIG. 7B is a flow chart of an uplink LSFC estimator that individuallyestimates the LSFC for each MS using a row of a diagonal matrix as pilotwith multiple pilot transmissions in a massive MIMO system.

FIG. 8 shows an exemplary schematic view of an uplink LSFC estimatorwith multiple pilot transmissions in a massive MIMO system.

FIG. 9 shows the MSE performance with respect to BS antenna number andSNR of the proposed uplink LSFC estimator without SSFC knowledge usingonly one training block.

FIG. 10 shows the MSE performance with respect to BS antenna number andSNR of the conventional single-block uplink LSFC estimator with perfectSSFC knowledge and the proposed uplink LSFC estimator without SSFCknowledge using multiple training blocks.

FIG. 11 is a flow chart of a method of estimating uplink LSFC inaccordance with one novel aspect.

FIG. 12 is an exemplary diagram illustrating a downlink MIMO system inaccordance with one novel aspect.

FIG. 13 is a flow chart and schematic view of a downlink semi-blind LSFCestimator that resides at each MS and estimates the LSFC of the MS usinga semi-unitary matrix as pilot in a massive MIMO system.

FIG. 14 is a flow chart and schematic view of a downlink blind LSFCestimator that resides at each MS and estimates the LSFC of the MSexploiting only the statistics of the unknown broadcast signal in amassive MIMO system.

FIG. 15A-15B show the MSE performance with respect to SNR and trainingperiod of the proposed downlink LSFC estimators without SSFC knowledge.

FIG. 16 is a flow chart of a method of estimating downlink LSFC inaccordance with one novel aspect.

DETAILED DESCRIPTION

Reference will now be made in detail to some embodiments of theinvention, examples of which are illustrated in the accompanyingdrawings.

Notation: (.)^(T), (.)^(H), (.)* represent the transpose, conjugatetranspose, and conjugate of the enclosed items, respectively, vec(.) isthe operator that forms one tall vector by stacking columns of theenclosed matrix, whereas Diag(.) translate a vector into a diagonalmatrix with the vector entries being the diagonal terms. While E{.},∥.∥, and ∥.∥_(F), denote the expectation, vector l₂-norm, and Frobeniusnorm of the enclosed items, respectively,

and ⊙ respectively denote the Kronecker and Hadamard product operator.Denoted by I_(L), 1_(L), and 0_(L) respectively, are the (L×L) identitymatrix, L-dimensional all-one and all-zero column vectors, whereas1_(L×S) and 0_(L×S) are the matrix counterparts of the latter two.Almost surely convergence is denoted by

$\overset{a.s.}{}$

and the Kronecker delta function denoted by

$\delta_{ij}\overset{def}{=}\left\{ {\begin{matrix}{0,} & {i \neq j} \\{1,} & {i = j}\end{matrix}.} \right.$

FIG. 1 illustrates simplified block diagrams of a base station and aplurality of mobile stations in a single-cell massive MU-MIMO system 100in accordance with one novel aspect. Massive MU-MIMO system 100comprises a base station BS 101 having an M-antenna array and Ksingle-antenna mobile stations MS #1 to MS #K, wherein M>>K. For amulti-cell uplink system, pilot contamination may become a seriousdesign concern in the worst case when the same pilot sequences (i.e.,the same pilot symbols are place at the same time-frequency locations)happen to be used simultaneously in several neighboring cells and areperfectly synchronized in both carrier and time. In practice, there arefrequency, phase, and timing offset between any pair of pilot signalsand the number of orthogonal pilots is often sufficient to serve mobileusers in multiple cells. Moreover, neighbor cells may use the same pilotsequence but the pilot symbols are located in non-overlappingtime-frequency units, hence a pilot sequence is more likely beinterfered by uncorrelated asynchronous data sequence whose impact isnot as serious as the worst case and can be mitigated by properinter-cell coordination, frequency planning and some interferencesuppression techniques. Throughout the present application, thediscussion will be focused on the single-cell narrowband scenario. Theproposed method, however, is not limited thereto.

In the example of FIG. 1, BS 101 comprises memory 102, a processor 103,a scheduler 104, a MIMO codec 105, a precoder/beamformer 106, a channelestimator 107, and a plurality of transceivers coupled to a plurality ofantennas. Similarly, each MS comprises memory, a processor, a MIMOcodec, a precoder/beamformer, a channel estimator, and a transceivercouple to an antenna. Each wireless device receives RF signals from theantenna, converts them to baseband signals and sends them to theprocessor. Each RF transceiver also converts received baseband signalsfrom the processor, converts them to RF signals, and sends out to theantenna. For example, processor 103 processes the received basebandsignals and invokes different functional modules to perform features inthe device. Memory 102 stores program instructions and data to controlthe operations of the device. The functional modules carry outembodiments of the current invention. The functional modules may beconfigured and implemented by hardware, firmware, software, or anycombination thereof.

FIG. 2 is an exemplary diagram illustrating an uplink MIMO system 200 inaccordance with one novel aspect. MIMO system 200 comprises a basestation BS 201 having M antennas, and K mobile stations MS 1 to MS K. Inthe uplink direction, each k^(th) MS transmits pilot training signalsp_(k) to be received by BS 201 via M antennas. We assume a narrowbandcommunication environment in which a transmitted signal suffers fromboth large-scale and small-scale fading. The large-scale fadingcoefficients (LSFCs) for each uplink channel is denoted as β_(k)'s,while the small-scale fading coefficients (SSFCs) for each uplinkchannel is denoted as h_(k)'s. The K uplink packets place their pilot oflength T at the same time-frequency location so that, without loss ofgenerality, the corresponding received signals, arranged in matrix form,Yε

^(M×T) at the BS can be expressed as:

$Y = {{{\sum\limits_{k = 1}^{K}\; {\sqrt{\beta_{k}}h_{k}p_{k}^{H}}} + N} = {{{HD}_{\beta}^{1/2}P} + N}}$

where

-   -   H=[h₁, . . . , h_(K)]ε        ^(M×K) contains the SSFCs that characterize the K uplink        channels, h_(k)=Φ_(k) ^(1/2){tilde over (h)}_(k), {tilde over        (h)}_(k)˜CN(0_(M),I_(M)), where Φ_(k) is the spatial correlation        matrix at the BS side with respect to the k^(th) user    -   D_(β)=Diag(β₁, . . . , β_(K)) contains the LSFCs that        characterize the K uplink channels, vector β=[(β₁, . . . ,        β_(K))^(T)] whose elements β_(k)=s_(k)d_(k) ^(−α) describes the        shadowing effect, parameterized by independent identically        distributed (i.i.d) s_(k)'s with log₁₀(s_(k))˜N(0,σ_(s) ²), and        the pathloss which depends on the distance between the BS and MS        d_(k), with α>0    -   P=[p₁, . . . , p_(K)]^(H)ε        ^(K×T) is the K×T matrix where M>> T≧K and p_(k) is the pilot        sequence sent by MS k and p_(j) ^(H)p_(k)=0, ∀j#k (orthogonal        pilot sequences)    -   N=[n_(ij)], n_(ij)˜CN(0,1) is the noise matrix whose entries are        distributed according to CN(0,1).

We invoke the assumption that independent users are relatively far apart(with respect to the wavelength) and the k^(th) uplink channel vector isindependent of the l^(th) vector, ∀l≠k. We assume that {tilde over(h)}_(k) are i.i.d. and the SSFC H remains constant during a pilotsequence period, i.e., the channel's coherence time is greater than T,while the LSFC β varies much slower.

Unlike most of the existing works that focus on the estimation of thecomposite channel matrix HD_(β) ^(1/2), or equivalently, ignore theLSFC, it is beneficial for system performance to know H and D_(β) ^(1/2)separately. Even though the decoupled treatment of LSFCs and SSFCs hasbeen seen recently, the assumption that the former is well known isusually made. In ordinary MIMO systems, MMSE or LS criterion cannot beused directly to jointly estimate LSFC and SSFC owing to their coupling,and EM algorithm is a feasible alternative. However, EM has highcomputational complexity and convergence is not guaranteed. Inaccordance with one novel aspect, a timely accurate LSFC estimator foruplink massive MIMO without the prior knowledge of SSFC is proposed.

FIG. 3 is a flow chart of an uplink LSFC estimator that estimates allaccessing MSs' LSFCs simultaneously in a massive MIMO system 200 of FIG.2. In step 311, each MS k transmits assigned UL pilot p_(k). In step312, the BS receives pilot signals transmitted from all K MSs, whichbecomes the received signals denoted as Y. In step 313, the BSvectorizes

${\frac{1}{M}Y^{H}Y} - {I_{T}.}$

Finally, in step 314, the BS derives an estimator of LSFC {circumflexover (β)} by multiplying it with Diag(∥p₁∥⁻⁴, . . . , ∥p_(k)∥⁻⁴)·((1_(T)^(T)

P)⊙(P*

1_(T) ^(T)). The derivation of LSFC {circumflex over (β)} is as follows:

$\begin{matrix}{{{{\frac{1}{M}Y^{H}Y} - I_{T}} = {{{\frac{2}{M}\Re \left\{ {P^{H}D_{\beta}^{1/2}H^{H}N} \right\}} + {P^{H}D_{\beta}P} + {P^{H}{D_{\beta}^{\frac{1}{2}}\left( {{\frac{1}{M}H^{H}H} - I_{K}} \right)}D_{\beta}^{\frac{1}{2}}P} + {\frac{1}{M}N^{H}N} - I_{T}}\mspace{20mu} \overset{a.s}{\rightarrow}{P^{H}D_{\beta}P}}}\mspace{20mu} {{{vec}\left( {{\frac{1}{M}Y^{H}Y} - I_{T}} \right)}\overset{a.s}{\rightarrow}{\left( {\left( {1_{T} \otimes P^{H}} \right) \odot \left( {P^{T} \otimes 1_{T}} \right)} \right)\beta}}{\hat{\beta} = {{{Diag}\left( {{p_{1}}^{- 4},\ldots \mspace{14mu},{p_{K}}^{- 4}} \right)} \cdot \left( {\left( {1_{T}^{T} \otimes P} \right) \odot \left( {P^{*} \otimes 1_{T}^{T}} \right)} \right) \cdot {{vec}\left( {{\frac{1}{M}Y^{H}Y} - I_{T}} \right)}}}} & (1)\end{matrix}$

where due to the large number of BS antennas M, the large sample size ofthe receive signal shows the following convergence:

H = [h₁, …  , h_(K)] = [h_(ik)], N = [n₁, …  , n_(T)] = [n_(ij)];${{H^{H}N}\overset{a.s.}{\rightarrow}\left. {0_{K \times T}\text{:}\mspace{14mu} {Because}\mspace{14mu} {when}\mspace{14mu} M}\rightarrow\infty \right.},{{h_{k}^{H}n_{j}} = {{{\sum\limits_{i = 1}^{M}\; {h_{ik}^{*}n_{ij}}} \approx {{M \cdot E}\left\{ {h_{ik}^{*}n_{ij}} \right\}}} = 0.}}$${{{\frac{1}{M}H^{H}H} - I_{K}}\overset{a.s.}{\rightarrow}\left. {0_{K \times K}\text{:}\mspace{14mu} {Because}\mspace{14mu} {when}\mspace{14mu} M}\rightarrow\infty \right.},{{\frac{1}{M}h_{k}^{H}h_{j}} = {{{\frac{1}{M}{\sum\limits_{i = 1}^{M}\; {h_{ik}^{*}h_{ij}}}} \approx {\frac{1}{M}E\left\{ {h_{ik}^{*}h_{ij}} \right\}}} = {{{{\delta_{kj}.\frac{1}{M}}N^{H}N} - I_{T}}\overset{a.s.}{\rightarrow}\left. {0_{T \times T}\text{:}\mspace{14mu} {Because}\mspace{14mu} {when}\mspace{14mu} M}\rightarrow\infty \right.}}},{{\frac{1}{M}n_{k}^{H}n_{j}} = {{{\frac{1}{M}{\sum\limits_{i = 1}^{M}\; {n_{ik}^{*}n_{ij}}}} \approx {\frac{1}{M}E\left\{ {n_{ik}^{*}n_{ij}} \right\}}} = {\delta_{kj}.}}}$

By exploiting the properties of massive MIMO, the proposed LFSCestimator has low computational complexity while outperform the onederived from EM algorithm. The proposed LFSC estimator is of lowcomplexity, as no matrix inversion is needed when orthogonal pilots areused and does not require any knowledge of SSFCs. Furthermore, theconfiguration of massive MIMO makes the estimator robust against noise.

FIG. 4A is a flow chart of an uplink LSFC estimator that estimates theLSFC for each accessing MS individually in a massive MIMO system 200 ofFIG. 2. In step 411, each MS k transmits assigned UL pilot p_(k). Instep 412, the BS receives pilot signals from all K MSs, denoted as Y. Instep 413, {circumflex over (β)}=[{circumflex over (β)}₁, . . . ,{circumflex over (β)}_(K)] is decoupled as, ∀k, and the BS derives anestimator of each LSFC {circumflex over (β)}_(k) for each uplink channelto be:

$\begin{matrix}{{\hat{\beta}}_{k} = \frac{{p_{k}^{H}Y^{H}{Yp}_{k}} - {M{p_{k}}^{2}}}{M{p_{k}}^{4}}} & (2)\end{matrix}$

FIG. 4B is a flow chart of an uplink LSFC estimator that estimates theLSFC for each accessing MS individually using a row of a diagonal matrixas pilot in a massive MIMO system 200 of FIG. 2. In step 421, each MS ktransmits assigned UL pilot p_(k). In the example of FIG. 4B, the pilotis chosen to be a row of diagonal matrix, i.e., P=Diag(s₁, . . . . ,s_(K)) and T=K, where each pilot sequence p_(k)=[0, . . . , 0, s_(k), 0,. . . 0]. In step 422, the BS receives pilot signals from all K MSs,denoted as Y, where Y=[y₁, . . . , y_(K)]. In step 423, the BS decouplesthe LSFC estimator {circumflex over (β)}_(k) for each uplink channel tobe:

$\begin{matrix}{{\hat{\beta}}_{k} = \frac{{y_{k}}^{2} - M}{{Ms}_{k}^{2}}} & (3)\end{matrix}$

This estimator coincides with our prediction that the instantaneousreceived signal strength minus the noise power, ∥y_(k)∥²−M, isapproximately equal to the strength of the desired signal and thusfairly reflects the gain provided by large-scale fading if it is dividedby M s_(k) ², the total power emitted by user k (s_(k) ²) times thenumber of copies received at the BS (M).

FIG. 5 shows an exemplary schematic view of an uplink LSFC estimator 501in a massive MIMO system 200 of FIG. 2. In the example of FIG. 5, a basestation having M antennas receives radio signal Y from K mobile stationsMSs, each MS k transmits a pilot sequence p_(k), and the noise varianceis σ². The UL LSFC estimator 501 is able to derive the LSFC {circumflexover (β)}_(k) for each uplink channel with low computational complexityand without prior knowledge of the small-scale fading coefficients.

FIG. 6 is a flow chart of an uplink LSFC estimator that estimates allaccessing MSs' LSFCs simultaneously with multiple pilot transmissions ina massive MIMO system 200 of FIG. 2. In step 611, each MS k transmitsassigned UL pilot p_(k) J times. The J-time pilot transmissions can beachieved in different ways. In one example, the MS may transmit thepilot p_(k) by repeating the transmission J times in time domain. Inanother example, the MS may transmit the pilot p_(k) by repeating thetransmission J times in frequency domain. Note that, in our example,although the ip_(k) remains the same during the J transmissions,different p_(k) can be used for each of the different transmissions. Instep 612, the BS receives pilot signals from all K MSs, denoted as Y₁, .. . , Y_(J), where Y_(i) is the i^(th) received signal block at the BS.In step 613, the BS vectorizes

${\frac{1}{MJ}{\sum\limits_{i = 1}^{J}\; {Y_{i}^{H}Y_{i}}}} - {\frac{1}{J}{I_{T}.}}$

Finally, in step 614, the BS derives an estimator of LSFC {circumflexover (B)} by multiplying it with Diag(∥p₁∥⁻⁴, . . . , ∥p_(K)∥⁻⁴)·((1_(T)^(T)

P)⊙(P*

1_(T) ^(T))). If the J coherent resource blocks on time-frequency domainin which the LSFCs remain constant are available, then we have:

$\begin{matrix}{\hat{\beta} = {{{Diag}\left( {{p_{1}}^{- 4},\ldots \mspace{14mu},{p_{K}}^{- 4}} \right)} \cdot \left( {\left( {1_{T}^{T} \otimes P} \right) \odot \left( {P^{*} \otimes 1_{T}^{T}} \right)} \right) \cdot {{vec}\left( {{\frac{1}{MJ}{\sum\limits_{i = 1}^{J}\; {Y_{i}^{H}Y_{i}}}} - {\frac{1}{J}I_{T}}} \right)}}} & (4)\end{matrix}$

FIG. 7A is a flow chart of an uplink LSFC estimator that individuallyestimates the LSFC for each MS with multiple pilot transmissions in amassive MIMO system 200 of FIG. 2. In step 711, each MS k transmitsassigned UL pilot p_(k) J times. In step 712, the BS receives pilotsignals from all K MSs, denoted as Y₁, . . . , Y_(J). In step 713,{circumflex over (β)}_(k) is decoupled from {circumflex over(β)}=[{circumflex over (β)}₁, . . . , {circumflex over (β)}_(K)], ∀k,and the BS derives an estimator of each LSFC {circumflex over (β)}_(k)for each uplink channel to be:

$\begin{matrix}{{\hat{\beta}}_{k} = \frac{{\sum\limits_{i = 1}^{J}\; {p_{k}^{H}Y_{i}^{H}Y_{i}p_{k}}} - {{MJ}{p_{k}}^{2}}}{{MJ}{p_{k}}^{4}}} & (5)\end{matrix}$

FIG. 7B is a flow chart of an uplink LSFC estimator that individuallyestimates the LSFC for each MS using a row of a diagonal matrix as pilotwith multiple pilot transmissions in a massive MIMO system 200 of FIG.2. In step 721, each MS k transmits assigned UL pilot p_(k) J times. Inthe example of FIG. 7B, the pilot is chosen to be a row of diagonalmatrix, i.e., P=Diag(s₁, . . . , s_(K)), where each pilot sequencep_(k)=[0, . . . , 0, s_(k), 0, . . . 0]. In step 722, for each of thei^(th) pilot transmission, the BS receives pilot signals from all K MSs,denoted as Y_(i), where Y_(i)=[y₁ ^((i)), . . . , y_(K) ^((i))]. In step723, the BS decouples the LSFC estimator {circumflex over (β)}_(k) foreach uplink channel to be:

$\begin{matrix}{{\hat{\beta}}_{k} = \frac{{\sum\limits_{i = 1}^{J}\; {y_{k}^{(i)}}^{2}} - {MJ}}{{MJs}_{k}^{2}}} & (6)\end{matrix}$

While the diagonal pilots give lower computational burden, therequirement that an MS needs to transmit all pilot power in a time slotto achieve the same performance shows a risk of disobeying the maximumuser output power constraint. The decision of a suitable uplink pilotpattern is a trade-off between the computational complexity and maximumuser output power. In one alternative example, a Hadamard matrix isadopted as the pilot pattern. A Hadamard matrix is a square matrix whoserows or columns are mutually orthogonal and of ±1 entries. It isconjectured that a Hadamard matrix or rows of it as the pilot matrix P,the computation effort can be reduced significantly due to the fact thatthe calculation of Yp_(k) in equation (2) or Y_(i)p_(k) in equation (5)involves only column additions and subtractions of Y/Y_(i).

FIG. 8 shows an exemplary schematic view of an uplink LSFC estimator 801with multiple pilot transmissions in a massive MIMO system 200 of FIG.2. In the example of FIG. 8, a base station BS having M antennasreceives radio signal Y₁, . . . , Y_(J) from K mobile stations MSs, eachMS k transmits a pilot sequence p_(k) J times, and the noise variance isσ². The UL LSFC estimator 801 is able to derive the LSFC {circumflexover (β)}_(k) for each uplink channel with low computational complexityand without prior knowledge of the small-scale fading coefficients.

FIG. 9 shows the MSE performance with respect to BS antenna number andSNR of the proposed uplink LSFC estimator without SSFC knowledge usingonly one training block (J=1). As shown in FIG. 9, the MSE performanceimproves as the number of antenna M increases, and as the SNR increases.

FIG. 10 shows the MSE performance with respect to BS antenna number andSNR of the conventional single-block uplink LSFC estimator with perfectSSFC knowledge and the proposed uplink LSFC estimator without SSFCknowledge using multiple training blocks. As shown in FIG. 10, the MSEperformance of the conventional single-block uplink LSFC estimator withperfect SSFC knowledge is the best, as depicted by the dashed-line.However, the MSE performance of the proposed uplink LSFC estimatorwithout SSFC knowledge improves as the number of antenna M increases,and as the number of training blocks J increases. Furthermore, theconfiguration of massive MIMO makes the estimator robust against noise.

FIG. 11 is a flow chart of a method of estimating uplink LSFC inaccordance with one novel aspect. In step 1101, a base station (BS)receives radio signals transmitted from K mobile stations (MSs) in amassive MIMO uplink channel where M>>K. In step 1102, the BS vectorizesthe received radio signals denoted as a matrix Yε

^(M×T), the transmitted radio signals are orthogonal pilot signalsdenoted as a matrix Pε

^(K×T) transmitted from the K MSs, and T≧K is the pilot signal length.In step 1103, the BS derives an estimator of large-scale fadingcoefficients (LSFCs) of the uplink channel without knowing small-scalefading coefficients (SSFCs) of the uplink channel. In step 1104, the BSreceives pilot signals that are transmitted for J times over coherentradio resource blocks from the K MS. In step 1105, the BS derives a moreaccurate estimator of the LSFCs of the uplink channel based on themultiple pilot transmissions. In step 1106, the BS calculateselement-wise expression of the LSFCs for each of the kth uplink channelbased on the LSFCs estimator.

FIG. 12 is an exemplary diagram illustrating a downlink MIMO system 1200in accordance with one novel aspect. MIMO system 1200 comprises a basestation BS 1201 having M antennas, and K mobile stations MS 1 to MS K.In the downlink direction, BS 1201 transmits downlink packets via someor all of its M antennas to be received by some or all of the K MSs. Weassume a narrowband communication environment in which a transmittedsignal suffers from both large-scale and small-scale fading. Thelarge-scale fading coefficients (LSFCs) for each downlink channel isdenoted as β_(k)'s, while the small-scale fading coefficients (SSFCs)for each downlink channel is denoted as g_(k)'s. The length-T downlinkpackets of different BS antennas are placed at the same time-frequencylocations so that, without loss of generality, the correspondingreceived samples, arranged in matrix form, X^(H)=[x₁, . . . , x_(K)]^(H)at MSs can be expressed as

X ^(H) =[x ₁ , . . . ,x _(K)]^(H) =D _(β) ^(1/2) G ^(H) Q+Z ^(H)

where

-   -   G=[g₁, . . . , g_(K)]ε        ^(M×K) contains the SSFCs that characterize the K downlink        channels, g_(k)=Φ_(k) ^(1/2){tilde over (g)}_(k), {tilde over        (g)}_(k)˜CN(0_(M),I_(M)), where Φ_(k) is the spatial correlation        matrix at the BS side with respect to the k^(th) user    -   D_(β)=Diag(β₁, . . . , β_(K)) contains the LSFCs that        characterize the K downlink channels, vector β=[β₁, . . . ,        β_(K)]^(T) whose elements β_(k)=s_(k)d_(k) ^(−α) describes the        shadowing effect, parameterized by independent identically        distributed (i.i.d) s_(k)'s with log₁₀(s_(k))˜N(0,σ_(s) ²), and        the pathloss which depends on the distance between the BS and MS        d_(k), with α>0    -   Q=[q₁, . . . ,q_(T)]ε        ^(M×T) is a M×T matrix where T≦M, which can be a pilot matrix        containing orthogonal columns q_(i) ^(H)q_(j)=0, ∀i≠j or a data        matrix intended to serving different MSs    -   Z=[z_(ij)], z_(ij)˜CN(0,1) is the noise matrix whose entries are        distributed according to CN(0,1).

We invoke the assumption that independent users are relatively far apart(with respect to the wavelength) and the k^(th) downlink channel vectoris independent of the l^(th) vector, ∀l≠k. We assume that {tilde over(g)}_(k) are i.i.d. and the SSFC G remains constant during a pilot/datasequence period, i.e., the channel's coherence time is greater than T,while the LSFC β varies much slower. In accordance with one novelaspect, several accurate LSFC estimators for downlink massive MIMOwithout the prior knowledge of SSFC are proposed. By exploiting theproperties of massive MIMO, it has low computational complexity.

FIG. 13 is a flow chart and schematic view of a downlink semi-blind LSFCestimator 1311 that resides at each MS and estimates the LSFC of the MSusing a semi-unitary matrix as pilot in a massive MIMO system 1200 ofFIG. 12. In step 1301, the BS transmits downlink pilot Q to the K MSs.In step 1302, the k^(th) MS receives signal x_(k) ^(H). In step 1303,the k^(th) MS recovers the LSFC {circumflex over (β)}_(k) for eachdownlink channel to be:

$\begin{matrix}{{\hat{\beta}}_{k} = \frac{{x_{k}}^{2} - T}{{Q}_{F}^{2}}} & (7)\end{matrix}$

where Q is a semi-unitary matrix and MS knows nothing but pilot power∥Q∥_(F) ².

In the embodiment of FIG. 13, let Q be a pilot matrix of the followingform with

$q_{t} = {\sqrt{P}\left\lbrack {0_{1 \times {({M - {TR} + r})}},u_{1\; t},0_{1 \times {({R - 1})}},u_{2\; t},0_{1 \times {({R - 1})}},\ldots \mspace{14mu},u_{Tt},0_{1 \times {({R - r - 1})}}} \right\rbrack}^{T}$  where   R = ⌊M/T⌋, r ≤ R − 1  U = [u_(ij)]  is  unitary  matrixQQ^(H) = P ⋅ Diag(0_(1 × (M − TR + r)), 1, 0_(1 × (R − 1)), 1, 0_(1 × (R − 1)), …  , 1, 0_(1 × (R − r − 1)))  Q^(H)Q = P ⋅ I_(T)$\mspace{20mu} {{{x_{k}}^{2} \approx {{\beta_{k}g_{k}^{H}{QQ}^{H}g_{k}} + {z_{k}}^{2}}}\overset{a.s.}{\rightarrow}{{{\beta_{k}{P \cdot T}} + {T\mspace{14mu} {if}\mspace{14mu} T}}1.}}$

In the example of FIG. 13, each k^(th) MS receives radio signal x_(k)from a BS having M antennas. The BS transmits a pilot signal denoted bymatrix Q, which is a semi-unitary matrix. The pilot power is ∥Q∥_(F) ²,and the noise variance is σ². The DL LSFC estimator 1311 is able toderive an estimate of the LSFC {circumflex over (β)}_(k) for eachdownlink channel with low computational complexity and without priorknowledge of the small-scale fading coefficients. Because in a massiveMIMO system, M≧T>> 1 gives

${{z_{k}^{H}Q^{H}g_{k}}\overset{a.s.}{\rightarrow}0},$

if T→∞, and ∥x_(k)∥²≈β_(k)g_(k) ^(H)g_(k)+∥z_(k)∥². In addition,

${\beta_{k}g_{k}^{H}{QQ}^{H}g_{k}} + {{{z_{k}}^{2}\overset{a.s.}{}\beta_{k}}{P \cdot T}} + T$

is because

${g_{k}^{H}{QQ}^{H}g_{k}} = {P\; {\sum\limits_{i = 1}^{T}{{{g_{{M - {{({T - i + 1})}R} + r + 1},k}}^{2}\overset{a.s.}{}{PT}}.}}}$

E{|g_(ik)|²}=PT if T→∞.

FIG. 14 is a flow chart and schematic view of a downlink blind LSFCestimator 1411 that resides at each MS and estimates the LSFC of the MSexploiting only the statistics of the unknown broadcast signal in amassive MIMO system 1200 of FIG. 12. In step 1401, the BS transmitsdownlink data signal Q=WD to a plurality of K′ MSs excluding MS 1 to K.In step 1402, the k^(th) MS receives signal x_(k) ^(H). In step 1403,the k^(th) MS recovers the LSFC for each downlink channel to be:

$\begin{matrix}{{\hat{\beta}}_{k} = \frac{{x_{k}}^{2} - T}{PT}} & (8)\end{matrix}$

where each MS using only statistics of unknown broadcast signal toestimate {circumflex over (β)}_(k).

In the embodiment of FIG. 14,

Q=WD

where

-   -   D=[d₁, . . . , d_(T)]ε        ^(K′×T): Data entries of D are unknown i.i.d. information        intended to K′ serving MSs excluding MS 1 to K. The power of D        entries is P/K′    -   W=[w₁, . . . , w_(K′)]ε        ^(M×K′): Unknown beamforming or precoding matrix, having        unit-norm columns, to those K′ serving MSs, and w_(i)        ^(H)w_(j)=δ_(ij)

$\begin{matrix}{{x_{k}}^{2} \approx {{\beta_{k}g_{k}^{H}{QQ}^{H}g_{k}} + {z_{k}}^{2}}} \\{= {{\beta_{k}{\overset{\sim}{g}}_{k}^{H}\Phi_{k}^{\frac{1}{2}}{QQ}^{H}\Phi_{k}^{\frac{1}{2}}{\overset{\sim}{g}}_{k}} + {{{z_{k}}^{2}\overset{a.s.}{}\beta_{k}}{{tr}\left( {\Phi_{k}^{\frac{1}{2}}{QQ}^{H}\Phi_{k}^{\frac{1}{2}}} \right)}} + T}} \\{= {{\beta_{k}{{tr}\left( {\Phi_{k}^{1/2}{WDD}^{H}W^{H}\Phi_{k}^{1/2}} \right)}}\; + T -}} \\{{{tr}\left( {\Phi_{k}^{1/2}{WDD}^{H}W^{H}\Phi_{k}^{1/2}} \right)}} \\{= {{tr}\left( {D^{H}W^{H}\Phi_{k}{WD}} \right)}} \\{\overset{def}{=}{{tr}\left( {D^{H}{AD}} \right)}} \\{= {{{{tr}\left( \begin{bmatrix}{d_{1}^{H}{Ad}_{1}} & \ldots & {d_{1}^{H}{Ad}_{T}} \\\vdots & \ddots & \vdots \\{d_{T}^{H}{Ad}_{1}} & \ldots & {d_{T}^{H}{Ad}_{T}}\end{bmatrix} \right)}\overset{a.s.}{}\frac{P}{K^{\prime}}}{{tr}\left( \begin{bmatrix}{trA} & \ldots & 0 \\\vdots & \ddots & \vdots \\0 & \ldots & {trA}\end{bmatrix} \right)}}} \\{= {\frac{PT}{K^{\prime}}{trA}}} \\{= {\frac{PT}{K^{\prime}}{{{tr}\left( \begin{bmatrix}{w_{1}^{H}\Phi_{k}w_{1}} & \ldots & {w_{1}^{H}\Phi_{k}w_{K^{\prime}}} \\\vdots & \ddots & \vdots \\{w_{K^{\prime}}^{H}\Phi_{k}w_{1}} & \ldots & {w_{K^{\prime}}^{H}\Phi_{k}w_{K^{\prime}}}\end{bmatrix} \right)}\overset{a.s.}{}}}} \\{{\frac{PT}{K^{\prime}M}{{tr}\left( \begin{bmatrix}{{tr}\; \Phi_{k}} & \ldots & 0 \\\vdots & \ddots & \vdots \\0 & \ldots & {{tr}\; \Phi_{k}}\end{bmatrix} \right)}}} \\{= {PT}}\end{matrix}$

In the example of FIG. 14, each k^(th) MS receives radio signal x_(k)from a BS having M antennas. The BS transmits a pilot signal denoted bymatrix Q=WD, the power is P, and the noise variance is σ². The DL LSFCestimator 1411 is able to derive the LSFC {circumflex over (β)}_(k) foreach downlink channel with low computational complexity and withoutprior knowledge of the small-scale fading coefficients. Note that whenN>> 1, two independent random vectors u,vε

^(N×1) has two properties: i)

${u^{H}{Au}} - {{{tr}(A)}\overset{a.s.}{}0}$

and ii)

$\frac{1}{M}u^{H}{{Av}\overset{a.s.}{}0.}$

Thus, with large dimensions of {tilde over (g)}_(k)'s, d_(i)'s,

${w_{i^{\prime}}s},{{\overset{\sim}{g}}_{k}^{H}\Phi_{k}^{\frac{1}{2}}{QQ}^{H}\Phi_{k}^{\frac{1}{2}}{{\overset{\sim}{g}}_{k}\overset{a.s.}{}{{tr}\left( {\Phi_{k}^{\frac{1}{2}}{QQ}^{H}\Phi_{k}^{\frac{1}{2}}} \right)}}},{\frac{K^{\prime}}{P}d_{i}^{H}{{{Ad}_{j}\overset{a.s.}{}{trA}} \cdot \delta_{ij}}},$

and

${M \cdot w_{i}^{H}}\Phi_{k}{w_{j}\overset{a.s.}{}{tr}}\; {\Phi_{k} \cdot {\delta_{ij}.}}$

FIG. 15A shows the MSE performance with respect to SNR and trainingperiod of the proposed downlink semi-blind LSFC estimators without SSFCknowledge. As shown in FIG. 15A, the MSE performance improves as the SNRincreases, and as the training period T increases.

FIG. 15B shows the MSE performance with respect to SNR and trainingperiod of the proposed downlink blind LSFC estimators without SSFCknowledge. As shown in FIG. 15B, the MSE performance improves as the SNRincreases, and as the training period T increases.

FIG. 16 is a flow chart of a method of estimating downlink LSFC inaccordance with one novel aspect. In step 1601, a mobile station (MS)receives radio signals transmitted from a base station (BS) having Mantennas in a massive MIMO system. The transmitted radio signals aredenoted as a matrix Q transmitted from the BS to K MS. In step 1602, theMS determines a received radio signal denoted as a vector x_(k) receivedby the MS that is the kth MS associated with a k^(th) downlink channel.In step 1603, the k^(th) MS derives an estimator of a large-scale fadingcoefficient (LSFC) of the k^(th) downlink channel without knowing asmall-scale fading coefficient (SSFC) of the kth downlink channel. Instep 1604, in a semi-blind LSFC estimation, matrix Q is a semi-unitarymatrix consisting of orthogonal pilot signals, and the LSFC of thek^(th) downlink channel is derived based on x_(k) and the transmittingpower of the pilot signals. In step 1605, in a blind LSFC estimation,matrix Q represents pre-coded data signals transmitted to K′ MS that aredifferent from the K MS. The LSFC of the k^(th) downlink channel isderived based on x_(k) and the transmitting power of the data signalswith unknown data information and unknown beamforming or precodinginformation.

Although the present invention has been described in connection withcertain specific embodiments for instructional purposes, the presentinvention is not limited thereto. Accordingly, various modifications,adaptations, and combinations of various features of the describedembodiments can be practiced without departing from the scope of theinvention as set forth in the claims.

What is claimed is:
 1. A method comprising: receiving radio signalstransmitted from K mobile stations (MSs) by a base station (BS) in amassive multiple input and multiple output (MIMO) uplink channel,wherein the BS has M antennas, and wherein M>>K; vectorizing receivedradio signals denoted as a matrix Yε

^(M×T), wherein the transmitted radio signals are orthogonal pilotsignals denoted as a matrix Pε

^(K×T) transmitted from the K MSs, and wherein T≧K is the pilot signallength; and deriving an estimator of large-scale fading coefficients(LSFCs) of the uplink channel with unknown small-scale fadingcoefficients (SSFCs) of the uplink channel.
 2. The method of claim 1,wherein the LSFCs are derived without iterative approach and withoutperforming matrix inversion.
 3. The method of claim 1, wherein the pilotsignals are denoted as P=[p₁, . . . , p_(K)]^(H), p_(k) represents ak^(th) pilot signal from a k^(th) MS to the BS via a k^(th) uplinkchannel, and wherein the LSFCs are derived to be$\hat{\beta} = {{{Diag}\left( {{p_{1}}^{- 4},\ldots \mspace{14mu},{p_{K}}^{- 4}} \right)} \cdot \left( {\left( {1_{T}^{T} \otimes P} \right) \odot \left( {P^{*} \otimes 1_{T}^{T}} \right)} \right) \cdot {{{vec}\left( {{\frac{1}{M}Y^{H}Y} - I_{T}} \right)}.}}$4. The method of claim 3, wherein the LSFCs are denoted as {circumflexover (β)}=[{circumflex over (β)}₁, . . . , {circumflex over (β)}_(K)],and wherein element-wise expressions render${\hat{\beta}}_{k} = \frac{{p_{k}^{H}Y^{H}{Yp}_{k}} - {M{p_{k}}^{2}}}{M{p_{k}}^{4}}$for the k^(th) uplink channel.
 5. The method of claim 3, wherein thepilot signals are transmitted for J times over coherent radio resourceblocks, and wherein the LSFCs are derived to be${\hat{\beta} = {{{Diag}\left( {{p_{1}}^{- 4},\ldots \mspace{14mu},{p_{K}}^{- 4}} \right)} \cdot \left( {\left( {1_{T}^{T} \otimes P} \right) \odot \left( {P^{*} \otimes 1_{T}^{T}} \right)} \right) \cdot {{vec}\left( {{\frac{1}{MJ}{\sum\limits_{i = 1}^{J}{Y_{i}^{H}Y_{i}}}} - {\frac{1}{J}I_{T}}} \right)}}},$where the received signals Y_(i) represents the i^(th) received radiosignals at the BS.
 6. The method of claim 5, wherein the LSFCs aredenoted as {circumflex over (β)}=[{circumflex over (β)}₁, . . . ,{circumflex over (β)}_(K)], and wherein element-wise expressions render${\hat{\beta}}_{k} = \frac{{\sum\limits_{i = 1}^{J}{p_{k}^{H}Y_{i}^{H}Y_{i}p_{k}}} - {{MJ}{p_{k}}^{2}}}{{MJ}{p_{k}}^{4}}$for the k^(th) uplink channel.
 7. The method of claim 5, wherein thepilot signals for different transmissions of the J times are different.8. The method of claim 1, wherein the pilot signals P is a diagonalmatrix and T=K.
 9. The method of claim 1, wherein the pilot signals P isa Hadamard matrix and T=K, wherein each of the rows or columns of P aremutually orthogonal and of ±1 entries.
 10. A method, comprising:receiving radio signals transmitted from a base station (BS) having Mantennas by a mobile station (MS) in a massive multiple input andmultiple output (MIMO) downlink system, wherein the transmitted radiosignals are denoted as a matrix Q transmitted from the BS to a pluralityof K MSs; determining a received radio signal denoted as vector x_(k)^(H) received by the MS that is the k^(th) MS associated with a k^(th)downlink channel; and deriving an estimator of large-scale fadingcoefficients (LSFC) of the k^(th) downlink channel with unknownsmall-scale fading coefficient (SSFC) of the k^(th) downlink channel.11. The method of claim 10, wherein the LSFC is derived withoutiterative approach and without performing matrix inversion.
 12. Themethod of claim 10, wherein matrix Q forms a semi-unitary matrix Q=[q₁,. . . , q_(T)]ε

^(M×T) of orthogonal pilot signals, wherein T is the length of eachpilot signal and T≦M, and wherein the pilot signals have a transmittingpower of ∥Q∥_(F) ².
 13. The method of claim 12, wherein the LSFC for thek^(th) downlink channel is derived to be${\hat{\beta}}_{k} = {\frac{{x_{k}}^{2} - T}{{Q}_{F}^{2}}.}$
 14. Themethod of claim 10, wherein matrix Q represents precoding data signalstransmitted to another plurality of K′ MSs that are different from the KMSs.
 15. The method of claim 14, wherein matrix Q=WD, wherein Wε

^(M×K′) represents an unknown precoding matrix and Dε

^(K′×T) represents unknown data information intended to the K′ MSs, andwherein T is the data signal length and 1<<T≦M.
 16. The method of claim15, wherein the LSFC for the kth downlink channel is derived to be${{\hat{\beta}}_{k} = \frac{{x_{k}}^{2} - T}{PT}},$ wherein Prepresents a transmit power of the precoding data signals.
 17. A mobilestation (MS), comprising: a receiver that receives radio signalstransmitted from a base station (BS) having M antennas in a massivemultiple input and multiple output (MIMO) downlink system, wherein thetransmitted radio signals are denoted as a matrix Q transmitted from theBS to a plurality of K MSs, and wherein a received radio signal denotedas vector x_(k) ^(H) is received by the MS that is the k^(th) MSassociated with a k^(th) downlink channel; and a channel estimationmodule that derives an estimator of large-scale fading coefficients(LSFC) of the k^(th) downlink channel with unknown small-scale fadingcoefficient (SSFC) of the k^(th) downlink channel.
 18. The MS of claim17, wherein the LSFC is derived without iterative approach and withoutperforming matrix inversion.
 19. The MS of claim 17, wherein matrix Qforms a semi-unitary matrix Q=[q₁, . . . , q_(T)]ε

^(M×T) of orthogonal pilot signals, wherein T is the length of eachpilot signal and 1<<T≦M, and wherein the pilot signals have atransmitting power of ∥Q∥_(F) ².
 20. The MS of claim 19, wherein theLSFC for the k^(th) downlink channel is derived to be${\hat{\beta}}_{k} = {\frac{{x_{k}}^{2} - T}{{Q}_{F}^{2}}.}$
 21. TheMS of claim 17, wherein matrix Q represents precoding data signalstransmitted to another plurality of K′ MSs that are different from the KMSs.
 22. The MS of claim 21, wherein matrix Q=WD, wherein Wε

^(M×K′) represents an unknown precoding matrix and Dε

^(K′×T) represents unknown data information intended to the K′ MSs, andwherein T is the data signal length and 1<<T≦M.
 23. The MS of claim 22,wherein the LSFC for the k^(th) downlink channel is derived to be${{\hat{\beta}}_{k} = \frac{{x_{k}}^{2} - T}{PT}},$ wherein Prepresents a transmit power of the precoding data signals.